Subgroups — #65372

Order 320.

120 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₅×C₄⋊Q₈	C₂	1	1	yes	maximal, characteristic, complemented
2	C₂×#688	C₂	1	2	yes	maximal
3	C₄×D₂₀	C₂	1	1	yes	maximal, characteristic
4	#3967	C₂	1	2	yes	maximal, complemented
5	C₄⋊#168	C₂	1	1	yes	maximal, characteristic, complemented
6	C₂×C₅×Q₈	C₂²	1	2	yes	—
7	C₂×D₂₀	C₂²	1	1	yes	characteristic
8	C₂×C₄×D₅	—	2	1	no	—
9	C₅×C₄⋊C₄	—	2	1	no	—
10	C₅×C₄⋊C₄	—	2	1	no	—
11	C₄²×C₅	C₂²	1	1	yes	characteristic
12	#688	—	2	4	no	—
13	#685	—	2	1	no	—
14	C₄⋊Dic₅	C₂²	1	1	yes	characteristic
15	C₂×#168	C₂²	1	2	yes	—
16	#461	—	5	1	no	maximal, Sylow
17	C₂²×D₅	—	2	1	no	—
18	C₅×Q₈	—	2	2	no	—
19	C₅×Q₈	—	2	2	no	—
20	C₂×C₄×C₅	—	2	2	no	—
21	C₂×C₄×C₅	D₄	1	1	yes	characteristic
22	C₂×C₄×C₅	D₄	1	1	yes	characteristic
23	C₂×C₄×C₅	C₂³	1	1	yes	characteristic
24	C₂×Dic₅	—	2	1	no	—
25	D₂₀	—	2	1	no	—
26	D₂₀	D₄	1	2	yes	—
27	C₄×D₅	—	4	2	no	—
28	#168	—	2	2	no	—
29	C₂×SD₁₆	—	5	2	no	—
30	C₄⋊Q₈	D₅	1	1	yes	characteristic, complemented
31	C₄×D₄	—	5	1	no	—
32	C₄⋊C₈	—	5	1	no	—
33	#103	—	5	2	no	—
34	C₂²×C₅	C₂×D₄	1	1	yes	characteristic
35	C₂×D₅	—	2	2	no	—
36	C₂×D₅	—	4	1	no	—
37	C₂×D₅	—	4	1	no	—
38	C₄×C₅	—	2	1	no	—
39	C₄×C₅	—	4	2	no	—
40	C₄×C₅	SD₁₆	1	2	yes	complemented
41	C₄×C₅	C₂×D₄	1	1	yes	characteristic
42	C₄×C₅	#41	1	1	yes	characteristic
43	Dic₅	—	4	1	no	—
44	C₂×Q₈	C₂×D₅	1	2	yes	—
45	C₂×D₄	—	5	1	no	—
46	C₂²×C₄	—	10	1	no	—
47	SD₁₆	—	10	4	no	—
48	C₂×C₈	—	5	2	no	—
49	C₄⋊C₄	—	2	1	no	—
50	C₄⋊C₄	—	2	1	no	—
51	C₄⋊C₄	—	5	1	no	—
52	C₂²⋊C₄	—	10	1	no	—
53	C₄²	C₂×D₅	1	1	yes	characteristic
54	C₂×C₅	#121	1	1	yes	characteristic
55	C₂×C₅	C₂×SD₁₆	1	1	yes	characteristic
56	C₂×C₅	#137	1	1	yes	characteristic
57	D₅	—	4	2	no	—
58	C₂³	—	10	1	no	—
59	Q₈	—	2	2	no	—
60	Q₈	—	2	2	no	—
61	D₄	—	5	2	no	—
62	D₄	—	10	1	no	—
63	C₂×C₄	—	2	2	no	—
64	C₂×C₄	—	10	1	no	—
65	C₂×C₄	—	20	2	no	—
66	C₂×C₄	C₂²⋊D₅	1	1	yes	characteristic
67	C₂×C₄	C₂²⋊D₅	1	1	yes	characteristic
68	C₂×C₄	C₂²×D₅	1	1	yes	characteristic
69	C₈	—	10	2	no	—
70	C₅	#461	1	1	yes	characteristic, complemented, Sylow
71	C₂²	—	10	2	no	—
72	C₂²	—	20	1	no	—
73	C₂²	—	20	1	no	—
74	C₂²	C₂×C₂²⋊D₅	1	1	yes	characteristic
75	C₄	—	2	1	no	—
76	C₄	—	4	2	no	—
77	C₄	—	20	1	no	—
78	C₄	#688	1	2	yes	complemented
79	C₄	D₄×D₅	1	1	yes	characteristic
80	C₄	#711	1	1	yes	characteristic
81	C₂	—	20	2	no	—
82	C₂	#4084	1	1	yes	characteristic
83	C₂	C₂×#688	1	1	yes	characteristic
84	C₂	#4088	1	1	yes	characteristic

	subgroup	G/H
center	C₂²	C₂×C₂²⋊D₅
commutator, cosocle	C₂×C₄×C₅	C₂³
Frattini	C₂×C₄	C₂²×D₅
Fitting	C₅×C₄⋊Q₈	C₂
socle	C₂²×C₅	C₂×D₄
prime radical	C₂×D₂₀	C₂²
hypercenter	C₄⋊Q₈	D₅
hypocenter	C₅	#461
LCS	G ▷ C₂×C₄×C₅ ▷ C₂×C₅ ▷ C₅
UCS	C₁ ◁ C₂² ◁ C₄² ◁ C₄⋊Q₈

Complements

Splittings G = N ⋊ H with N normal; H ≅ G/N.

N	H ≅ G/N	type
1. C₅×C₄⋊Q₈	C₂	semidirect
4. #3967	C₂	semidirect
5. C₄⋊#168	C₂	semidirect
30. C₄⋊Q₈	D₅	semidirect
40. C₄×C₅	SD₁₆	semidirect
70. C₅	#461	semidirect
78. C₄	#688	semidirect

Maximal subgroups

By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.

subgroup	index
C₅×C₄⋊Q₈	2
C₂×#688	2
C₄×D₂₀	2
#3967	2
C₄⋊#168	2
#461	5